Linear Algebra Examples

$p^{2} + 2 q^{2} = 11$

Step 1

Subtract $2 q^{2}$ from both sides of the equation.

$p^{2} = 11 - 2 q^{2}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{11 - 2 q^{2}}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$p = \sqrt{11 - 2 q^{2}}$

Step 3.2

Next, use the negative value of the $\pm$ to find the second solution.

$p = - \sqrt{11 - 2 q^{2}}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

$p = \sqrt{11 - 2 q^{2}}$

$p = - \sqrt{11 - 2 q^{2}}$

$p = \sqrt{11 - 2 q^{2}}$

$p = - \sqrt{11 - 2 q^{2}}$

Step 4

Set the radicand in $\sqrt{11 - 2 q^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$11 - 2 q^{2} \geq 0$

Step 5

Solve for $q$ .

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Step 5.1

Subtract $11$ from both sides of the inequality.

$- 2 q^{2} \geq - 11$

Step 5.2

Divide each term in $- 2 q^{2} \geq - 11$ by $- 2$ and simplify.

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Step 5.2.1

Divide each term in $- 2 q^{2} \geq - 11$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 q^{2}}{- 2} \leq \frac{- 11}{- 2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 2 q^{2}}{- 2} \leq \frac{- 11}{- 2}$

Step 5.2.2.1.2

Divide $q^{2}$ by $1$ .

$q^{2} \leq \frac{- 11}{- 2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

$q^{2} \leq \frac{11}{2}$

Step 5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{q^{2}} \leq \sqrt{\frac{11}{2}}$

Step 5.4

Simplify the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Pull terms out from under the radical.

$| q | \leq \sqrt{\frac{11}{2}}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\sqrt{\frac{11}{2}}$ .

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Step 5.4.2.1.1

Rewrite $\sqrt{\frac{11}{2}}$ as $\frac{\sqrt{11}}{\sqrt{2}}$ .

$| q | \leq \frac{\sqrt{11}}{\sqrt{2}}$

Step 5.4.2.1.2

Multiply $\frac{\sqrt{11}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| q | \leq \frac{\sqrt{11}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.4.2.1.3

Combine and simplify the denominator.

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Step 5.4.2.1.3.1

Multiply $\frac{\sqrt{11}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.4.2.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.4.2.1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.4.2.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.4.2.1.3.5

Add $1$ and $1$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.4.2.1.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.4.2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.4.2.1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.4.2.1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 5.4.2.1.3.6.4.1

Cancel the common factor.

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.4.2.1.3.6.4.2

Rewrite the expression.

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{2^{1}}$

Step 5.4.2.1.3.6.5

Evaluate the exponent.

$| q | \leq \frac{\sqrt{11} \sqrt{2}}{2}$

Step 5.4.2.1.4

Simplify the numerator.

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Step 5.4.2.1.4.1

Combine using the product rule for radicals.

$| q | \leq \frac{\sqrt{11 \cdot 2}}{2}$

Step 5.4.2.1.4.2

Multiply $11$ by $2$ .

$| q | \leq \frac{\sqrt{22}}{2}$

Step 5.5

Write $| q | \leq \frac{\sqrt{22}}{2}$ as a piecewise.

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Step 5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$q \geq 0$

Step 5.5.2

In the piece where $q$ is non-negative, remove the absolute value.

$q \leq \frac{\sqrt{22}}{2}$

Step 5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$q < 0$

Step 5.5.4

In the piece where $q$ is negative, remove the absolute value and multiply by $- 1$ .

$- q \leq \frac{\sqrt{22}}{2}$

Step 5.5.5

Write as a piecewise.

${\begin{cases} q \leq \frac{\sqrt{22}}{2} & q \geq 0 \\ - q \leq \frac{\sqrt{22}}{2} & q < 0 \end{cases}$

Step 5.6

Find the intersection of $q \leq \frac{\sqrt{22}}{2}$ and $q \geq 0$ .

$0 \leq q \leq \frac{\sqrt{22}}{2}$

Step 5.7

Solve $- q \leq \frac{\sqrt{22}}{2}$ when $q < 0$ .

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Step 5.7.1

Divide each term in $- q \leq \frac{\sqrt{22}}{2}$ by $- 1$ and simplify.

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Step 5.7.1.1

Divide each term in $- q \leq \frac{\sqrt{22}}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- q}{- 1} \geq \frac{\frac{\sqrt{22}}{2}}{- 1}$

Step 5.7.1.2

Simplify the left side.

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Step 5.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{q}{1} \geq \frac{\frac{\sqrt{22}}{2}}{- 1}$

Step 5.7.1.2.2

Divide $q$ by $1$ .

$q \geq \frac{\frac{\sqrt{22}}{2}}{- 1}$

Step 5.7.1.3

Simplify the right side.

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Step 5.7.1.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt{22}}{2}}{- 1}$ .

$q \geq - 1 \cdot \frac{\sqrt{22}}{2}$

Step 5.7.1.3.2

Rewrite $- 1 \cdot \frac{\sqrt{22}}{2}$ as $- \frac{\sqrt{22}}{2}$ .

$q \geq - \frac{\sqrt{22}}{2}$

Step 5.7.2

Find the intersection of $q \geq - \frac{\sqrt{22}}{2}$ and $q < 0$ .

$- \frac{\sqrt{22}}{2} \leq q < 0$

Step 5.8

Find the union of the solutions.

$- \frac{\sqrt{22}}{2} \leq q \leq \frac{\sqrt{22}}{2}$

Step 6

The domain is all values of $q$ that make the expression defined.

Interval Notation:

$[- \frac{\sqrt{22}}{2}, \frac{\sqrt{22}}{2}]$

Set-Builder Notation:

${q | - \frac{\sqrt{22}}{2} \leq q \leq \frac{\sqrt{22}}{2}}$

Step 7